Optimal. Leaf size=220 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 a^{3/2} d}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{5 a^2 d}-\frac{3 \cot (c+d x)}{128 a d \sqrt{a \sin (c+d x)+a}}+\frac{19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.875979, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2880, 2772, 2773, 206, 3044, 2980} \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{128 a^{3/2} d}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{5 a^2 d}-\frac{3 \cot (c+d x)}{128 a d \sqrt{a \sin (c+d x)+a}}+\frac{19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2880
Rule 2772
Rule 2773
Rule 206
Rule 3044
Rule 2980
Rubi steps
\begin{align*} \int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac{\int \csc ^6(c+d x) \sqrt{a+a \sin (c+d x)} \left (1+\sin ^2(c+d x)\right ) \, dx}{a^2}-\frac{2 \int \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{a^2}\\ &=\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 a^2 d}+\frac{\int \csc ^5(c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{a}{2}+\frac{17}{2} a \sin (c+d x)\right ) \, dx}{5 a^3}-\frac{7 \int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{4 a^2}\\ &=\frac{7 \cot (c+d x) \csc ^2(c+d x)}{12 a d \sqrt{a+a \sin (c+d x)}}+\frac{19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 a^2 d}-\frac{35 \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{24 a^2}+\frac{143 \int \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{80 a^2}\\ &=\frac{35 \cot (c+d x) \csc (c+d x)}{48 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 a^2 d}-\frac{35 \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{32 a^2}+\frac{143 \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{96 a^2}\\ &=\frac{35 \cot (c+d x)}{32 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{64 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 a^2 d}-\frac{35 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{64 a^2}+\frac{143 \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{128 a^2}\\ &=-\frac{3 \cot (c+d x)}{128 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{64 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 a^2 d}+\frac{143 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{256 a^2}+\frac{35 \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{32 a d}\\ &=\frac{35 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{32 a^{3/2} d}-\frac{3 \cot (c+d x)}{128 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{64 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 a^2 d}-\frac{143 \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{128 a d}\\ &=-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{128 a^{3/2} d}-\frac{3 \cot (c+d x)}{128 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{64 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)}}{5 a^2 d}\\ \end{align*}
Mathematica [A] time = 1.36168, size = 412, normalized size = 1.87 \[ -\frac{\csc ^{15}\left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3 \left (-7100 \sin \left (\frac{1}{2} (c+d x)\right )-2880 \sin \left (\frac{3}{2} (c+d x)\right )+144 \sin \left (\frac{5}{2} (c+d x)\right )-10 \sin \left (\frac{7}{2} (c+d x)\right )-30 \sin \left (\frac{9}{2} (c+d x)\right )+7100 \cos \left (\frac{1}{2} (c+d x)\right )-2880 \cos \left (\frac{3}{2} (c+d x)\right )-144 \cos \left (\frac{5}{2} (c+d x)\right )-10 \cos \left (\frac{7}{2} (c+d x)\right )+30 \cos \left (\frac{9}{2} (c+d x)\right )+150 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-150 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-75 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+75 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+15 \sin (5 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-15 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{640 d (a (\sin (c+d x)+1))^{3/2} \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.076, size = 180, normalized size = 0.8 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{640\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 15\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{9/2}{a}^{7/2}-70\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{9/2}+128\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{11/2}+15\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}+70\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{13/2}-15\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{15/2} \right ){a}^{-{\frac{19}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.23256, size = 1301, normalized size = 5.91 \begin{align*} \frac{15 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a} - 9 \, a \cos \left (d x + c\right ) +{\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \,{\left (15 \, \cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} - 38 \, \cos \left (d x + c\right )^{3} - 194 \, \cos \left (d x + c\right )^{2} -{\left (15 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} - 28 \, \cos \left (d x + c\right )^{2} + 166 \, \cos \left (d x + c\right ) + 317\right )} \sin \left (d x + c\right ) + 151 \, \cos \left (d x + c\right ) + 317\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{2560 \,{\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d -{\left (a^{2} d \cos \left (d x + c\right )^{5} + a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.58785, size = 1091, normalized size = 4.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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